1. Deconfined quantum criticality
Science 303, 1490 (2004); Physical Review B 70, (2004), 71, and 71, (2005), cond-mat/
Leon Balents (UCSB) Lorenz Bartosch (Harvard) Anton Burkov (Harvard) Matthew Fisher (UCSB) Subir Sachdev (Harvard) Krishnendu Sengupta (HRI, India) T. Senthil (MIT and IISc) Ashvin Vishwanath (Berkeley)
Talk online at

2. Outline
Magnetic quantum phase transitions in “dimerized” Mott insulators: Landau-Ginzburg-Wilson (LGW) theory
Magnetic quantum phase transitions of Mott insulators on the square lattice A. Breakdown of LGW theory B. Berry phases C. Spinor formulation and deconfined criticality

3. I. Magnetic quantum phase transitions in “dimerized” Mott insulators:
Landau-Ginzburg-Wilson (LGW) theory:
Second-order phase transitions described by fluctuations of an order parameter associated with a broken symmetry

4. M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist, cond-mat/0309440.
TlCuCl3
M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist, cond-mat/

5. Coupled Dimer Antiferromagnet
M. P. Gelfand, R. R. P. Singh, and D. A. Huse, Phys. Rev. B 40, (1989).
N. Katoh and M. Imada, J. Phys. Soc. Jpn. 63, 4529 (1994).
J. Tworzydlo, O. Y. Osman, C. N. A. van Duin, J. Zaanen, Phys. Rev. B 59, 115 (1999).
M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, (2002).
S=1/2 spins on coupled dimers

7. Weakly coupled dimers

8. Weakly coupled dimers
Paramagnetic ground state

9. Weakly coupled dimers
Excitation:
S=1 quasipartcle

10. Weakly coupled dimers
Excitation:
S=1 quasipartcle

11. Weakly coupled dimers
Excitation:
S=1 quasipartcle

12. Weakly coupled dimers
Excitation:
S=1 quasipartcle

13. Weakly coupled dimers
Excitation:
S=1 quasipartcle

14. Weakly coupled dimers Excitation: S=1 quasipartcle
Energy dispersion away from
antiferromagnetic wavevector

15. TlCuCl3 S=1 quasi-particle
N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B (2001).

16. Coupled Dimer Antiferromagnet

17. close to 1 l
Weakly dimerized square lattice

18. l close to 1 Weakly dimerized square lattice Excitations:
2 spin waves (magnons)
Ground state has long-range spin density wave (Néel) order at wavevector K= (p,p)

19. TlCuCl3
J. Phys. Soc. Jpn 72, 1026 (2003)

20. lc = 0. 52337(3) M. Matsumoto, C. Yasuda, S. Todo, and H
lc = (3) M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, (2002)
T=0
Néel state
Quantum paramagnet 1
Pressure in TlCuCl3
The method of bond operators (S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990)) provides a quantitative description of spin excitations in TlCuCl3 across the quantum phase transition (M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist, Phys. Rev. Lett. 89, (2002))

21. LGW theory for quantum criticality
S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)

22. LGW theory for quantum criticality
S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)
A.V. Chubukov, S. Sachdev, and J.Ye, Phys. Rev. B 49, (1994)

23. Outline
Magnetic quantum phase transitions in “dimerized” Mott insulators: Landau-Ginzburg-Wilson (LGW) theory
Magnetic quantum phase transitions of Mott insulators on the square lattice A. Breakdown of LGW theory B. Berry phases C. Spinor formulation and deconfined criticality

24. A. Breakdown of LGW theory
II. Magnetic quantum phase transitions of Mott insulators on the square lattice:
A. Breakdown of LGW theory

25. Ground state has long-range Néel order
Square lattice antiferromagnet
Ground state has long-range Néel order

26. Square lattice antiferromagnet
Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.

27. Square lattice antiferromagnet
Destroy Neel order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.

28. LGW theory for quantum criticality

29. Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries

30. “Liquid” of valence bonds has fractionalized S=1/2 excitations
Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has fractionalized S=1/2 excitations

31. “Liquid” of valence bonds has fractionalized S=1/2 excitations
Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has fractionalized S=1/2 excitations

32. “Liquid” of valence bonds has fractionalized S=1/2 excitations
Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has fractionalized S=1/2 excitations

33. “Liquid” of valence bonds has fractionalized S=1/2 excitations
Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has fractionalized S=1/2 excitations

34. “Liquid” of valence bonds has fractionalized S=1/2 excitations
Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has fractionalized S=1/2 excitations

35. “Liquid” of valence bonds has fractionalized S=1/2 excitations
Problem: there is no state with a gapped, stable S=1 quasiparticle and no broken symmetries
“Liquid” of valence bonds has fractionalized S=1/2 excitations

47. The VBS state does have a stable S=1 quasiparticle excitation

48. The VBS state does have a stable S=1 quasiparticle excitation

49. The VBS state does have a stable S=1 quasiparticle excitation

50. The VBS state does have a stable S=1 quasiparticle excitation

51. The VBS state does have a stable S=1 quasiparticle excitation

52. The VBS state does have a stable S=1 quasiparticle excitation

53. LGW theory of multiple order parameters
Distinct symmetries of order parameters permit couplings only between their energy densities

54. LGW theory of multiple order parameters
First order transition g g g

55. LGW theory of multiple order parameters
First order transition g g g

56. Outline
Magnetic quantum phase transitions in “dimerized” Mott insulators: Landau-Ginzburg-Wilson (LGW) theory
Magnetic quantum phase transitions of Mott insulators on the square lattice A. Breakdown of LGW theory B. Berry phases C. Spinor formulation and deconfined criticality

57. II. Magnetic quantum phase transitions of Mott insulators on the square lattice:
B. Berry phases

58. Quantum theory for destruction of Neel order
Ingredient missing from LGW theory: Spin Berry Phases

59. Quantum theory for destruction of Neel order
Ingredient missing from LGW theory: Spin Berry Phases

60. Quantum theory for destruction of Neel order

61. Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a

62. Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a

63. Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a

64. Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a

65. Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a

66. Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a

67. Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a
Change in choice of is like a “gauge transformation”

68. Quantum theory for destruction of Neel order
Discretize imaginary time: path integral is over fields on the sites of a cubic lattice of points a
Change in choice of is like a “gauge transformation”
The area of the triangle is uncertain modulo 4p, and the action has to be invariant under

69. Quantum theory for destruction of Neel order
Ingredient missing from LGW theory: Spin Berry Phases
Sum of Berry phases of all spins on the square lattice.

70. Quantum theory for destruction of Neel order
Partition function on cubic lattice
LGW theory: weights in partition function are those of a classical ferromagnet at a “temperature” g

71. Quantum theory for destruction of Neel order
Partition function on cubic lattice
Modulus of weights in partition function: those of a classical ferromagnet at a “temperature” g
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

72. Outline
Magnetic quantum phase transitions in “dimerized” Mott insulators: Landau-Ginzburg-Wilson (LGW) theory
Magnetic quantum phase transitions of Mott insulators on the square lattice A. Breakdown of LGW theory B. Berry phases C. Spinor formulation and deconfined criticality

73. C. Spinor formulation and deconfined criticality
II. Magnetic quantum phase transitions of Mott insulators on the square lattice:
C. Spinor formulation and deconfined criticality

74. Quantum theory for destruction of Neel order
Partition function on cubic lattice
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

75. Quantum theory for destruction of Neel order
Partition function on cubic lattice
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

76. Quantum theory for destruction of Neel order
Partition function on cubic lattice
Partition function expressed as a gauge theory of spinor degrees of freedom
S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002)

77. Large g effective action for the Aam after integrating zam
This theory can be reliably analyzed by a duality mapping.
The gauge theory is in a confining phase, and there is VBS order in the ground state. (Proliferation of monopoles in the presence of Berry phases).
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).
S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990).
K. Park and S. Sachdev, Phys. Rev. B 65, (2002).

78. or g

79. Ordering by quantum fluctuations

80. Ordering by quantum fluctuations

81. Ordering by quantum fluctuations

82. Ordering by quantum fluctuations

83. Ordering by quantum fluctuations

84. Ordering by quantum fluctuations

85. Ordering by quantum fluctuations

86. Ordering by quantum fluctuations

87. Ordering by quantum fluctuations

88. ? or g

89. Theory of a second-order quantum phase transition between Neel and VBS phases
Second-order critical point described by emergent fractionalized degrees of freedom (Am and za );
Order parameters (j and Yvbs ) are “composites” and of secondary importance
S. Sachdev and R. Jalabert, Mod. Phys. Lett. B 4, 1043 (1990); G. Murthy and S. Sachdev, Nuclear Physics B 344, 557 (1990); C. Lannert, M.P.A. Fisher, and T. Senthil, Phys. Rev. B 63, (2001); S. Sachdev and K. Park, Annals of Physics, 298, 58 (2002); O. Motrunich and A. Vishwanath, Phys. Rev. B 70, (2004)
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

90. Phase diagram of S=1/2 square lattice antiferromagnetor g
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

91. Conclusions
New quantum phases induced by Berry phases: VBS order in the antiferromagnet
Critical resonating-valence-bond states describes the quantum phase transition from the Neel to the VBS
Emergent gauge fields are essential for a full description of the low energy physics.